SOLUTION: given the function f(x)=|x|-|x+3|. (a) Convert f(x) to a piecewise-defined function on the fundamental definition of |x|. (b). Graph the function. (c) Find the x- and y-intercep

Algebra ->  Angles -> SOLUTION: given the function f(x)=|x|-|x+3|. (a) Convert f(x) to a piecewise-defined function on the fundamental definition of |x|. (b). Graph the function. (c) Find the x- and y-intercep      Log On


   

Question 1108345: given the function f(x)=|x|-|x+3|.
(a) Convert f(x) to a piecewise-defined function on the fundamental definition of |x|.
(b). Graph the function.
(c) Find the x- and y-intercepts of the graph.
(d) Find the zeros of the function.
Answer by greenestamps(13214) About Me  (Show Source):
You can put this solution on YOUR website!


The graph of f%28x%29=abs%28x%29 has its vertex at (0,0); the graph of f%28x%29+=+abs%28x%2B3%29 has its vertex at (-3,0). The x coordinates of the vertices determine where the function must be separated into pieces.

So we will have one function for x less than -3, another for x greater than or equal to -3 and less than 0, and a third for x greater than or equal to 0.

x < -3: abs%28x%29+=+-x; abs%28x%2B3%29+=+-x-3; abs%28x%29-abs%28x%2B3%29+=+%28-x%29+-+%28-x-3%29+=+3

-3 <= x < 0: abs%28x%29+=+-x; abs%28x%2B3%29+=+x%2B3' abs%28x%29-abs%28x%2B3%29+=+%28-x%29+=+%28x%2B3%29+=+-2x-3

x >= 0: abs%28x%29+=+x; abs%28x%2B3%29+=+x%2B3; abs%28x%29-abs%28x%2B3%29+=+%28x%29+=+%28x%2B3%29+=+-3

The piecewise function is
f%28x%29+=+3 for x < -3;
f%28x%29+=+-2x-3 for 3 <= x < 0;
f%28x%29+=+-3 for x >= 0

The graph:

graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cabs%28x%29-abs%28x%2B3%29%29

The y intercept is when x is 0: (0,-3).

The x intercept(s) are when y is 0. Either algebraically or from the graph, the only x intercept is (-1.5,0).